GATE EC 2015 Set 3 — Question 27

Let the LTI system have an impulse response $h(t)$.
Let the input to the system be $x(t)$ and the output be $y(t)$. The relationship is given by the convolution integral: $y(t) = x(t) * h(t)$. $h(t) = \delta(t) * h(t)$. $s(t) = u(t) * h(t) = \int_{-\infty}^{\infty} u(t-\tau) h(\tau) d\tau = \int_{-\infty}^{t} h(\tau) d\tau$.
This equation shows that the unit step response is the integral of the impulse response.
To find the impulse response $h(t)$ from the step response $s(t)$, we can differentiate the above equation with respect to $t$:
$\frac{d}{dt}s(t) = \frac{d}{dt} \left( \int_{-\infty}^{t} h(\tau) d\tau \right) = h(t)$.
Thus, the impulse response is obtained by differentiating the unit step response. This corresponds to option (B).Let's analyze the other options for completeness: We know that the unit ramp is the integral of the unit step function: $ramp(t) = \int_{-\infty}^{t} u(\tau) d\tau$.
Also, the unit step is the derivative of the unit ramp: $u(t) = \frac{d}{dt} ramp(t)$.
The ramp response is $r(t) = ramp(t) * h(t)$.
Differentiating the ramp response:
$\frac{d}{dt}r(t) = \frac{d}{dt} [ramp(t) * h(t)] = [\frac{d}{dt} ramp(t)] * h(t) = u(t) * h(t) = s(t)$.
So, differentiating the unit ramp response gives the unit step response, not the impulse response. Option (A) is incorrect. $\int_{-\infty}^{t} s(\tau) d\tau = \int_{-\infty}^{t} [u(\tau) * h(\tau)] d\tau = [\int_{-\infty}^{t} u(\tau) d\tau] * h(\tau) = ramp(t) * h(t) = r(t)$.
So, integrating the unit step response gives the unit ramp response. Option (D) is incorrect.Therefore, the correct statement is that the impulse response of an LTI system can be obtained by differentiating the unit step response.